1. Field of the Invention
The present invention relates to a chaos detection system, more particularly to a chaos processor which is able to determine whether an input signal is obtained from a random noise or from a meaningful information containing qualitative features of the strange attractor.
2. Description of the Prior Art
Recently, there have been active studies for seeking a process for estimating the future movements of the nature phenomena, such as the flow of water, air, and blood etc., the movement may be predetermined by the mathematical regularity of the movements gained.
Dynamics system can be defined as a system in which its states are varying with respect to time.
The dynamics system is called a stable system when the steady state solution remains in one point, which is in turn called an equilibrium point. When the attractor of the system makes a closed loop, the system is called a periodic system. When the attractor has a shape of doughnut, it is called a quasi-periodic system.
The procedure for obtaining the attractor of the dynamics system will be described as follows.
Generally, an nth order dynamics system have n state equations, and the state equations indicate the variation ratio of the states of the dynamics system against the variation of time as shown in equation (1). EQU dx1/dt=f(x1, x2, . . . , xn), . . . dxn/dt=f(x1, x2, . . . , xn)(1)
where, f: R.sup.N .fwdarw.R.sup.N stands for a nonlinear mapping, and x1, x2, . . . , xn stands for states respectively.
Hereinafter, a pendulum motion will be described as an elemental example of the dynamics system, the pendulum motion is expressed by 2 state equations in that the pendulum motion is a second order dynamics system as follows.
dx1/dt=f(x1, x2) PA1 dx2/dt=f(x1, x2)
The solutions of the above state equations consist of a transient solution and a steady state solution. The steady state solution can be expressed in a state space, in which each state variable makes an axis of the state space, so as to express the steady state solution entirely.
Namely, the steady state solution at a given time can be expressed as a point in the state space. A set of the points presented in the state space is called an attractor of dynamics system.
If the given dynamics system has a finite state, i.e., a finite nth order dynamics system, the system has a four forms of the attractor. The dynamics systems are classified into four types according to the types of the attractor of the dynamics system.
Namely, the steady linear system which is the most simple dynamics system has one point attractor in the state space, which is called an equilibrium point. Also, the dynamics system having the steady state solution, and the solution being a periodic solution, has a closed loop-shaped attractor in the state space, which is then called a limit cycle.
And, the dynamics system having kth order subharmonic solution, which has a k periods, has a doughnut-shaped attractor. The doughnut is called a torus.
The attractor except those of the above-mentioned dynamics system is a strange attractor, and this type of dynamics system is called a chaos system.
Namely, the chaos system refers to a system having a strange attractor in the state space, with the exception of said one point attractor, said limit cycle, and said torus.
As mentioned above, the attractor may be constructed from the state equation which is presenting the state of the movements of the nature phenomena. In that case, all the n state equations are known in the nth order dynamics system, the attractor may be constructed easily.
In fact, however, it is practically impossible to access the whole n state equations let alone state variables, in a given nth order dynamics system. Accordingly, the endeavor has been devoted to construct the attractor of nth order dynamics system from only one state variable.
Namely, when an attractor is obtained from the steady state solution of a given state variable, the attractor may be presented in the state space.
Since the desired attractor may not be gained in the state space, an embedding space should be introduced.
As described above, the constructing of the attractor of the nth dynamics system from a given state variable is called an attractor reconstruction. The attractor reconstruction plays an important role among the researchers who are dependent upon the experiments.
The attractor reconstruction has been proposed by Floris Tarkens in the mid of 1980's.
And, the trace time is divided into the same periods, and the corresponding state value of the divided time is presented as an embedding vector g(t). The vector g(t) is satisfied with the following equation. EQU g(t)={y(t), y(t+.tau.), . . . , y(t+n.tau.)}
where, y(t) stands for a state value, .tau. is a delay time which is divided into the same periods, and n+1 is an embedding dimension.
If the delay time and the embedding dimension are fixed, the embedding vector is expressed as one point. And the delay time and the embedding dimension is altered, then the embedding vector draws a trace in the embedding space.
The embedding vector trace in the given nth order embedding space may not exactly the same as the trace of the attractor of the given dynamics system, but the embedding vector trace has relation with the trace of the original dynamics system in the qualitative viewpoint (pattern face).
However, it needs to determine whether the attractor is constructed from a random noise or the meaningful information from chaos system.
There are two kinds of methods for analyzing the reconstructed attractor, one is to analyze the qualitative feature of the reconstructed attractor, which is called a qualitative method, and the other is to analyze the degree of the pattern such as a slope of the attractor, which is called a quantitative method.
In case that said reconstructed attractor has a pattern of an equilibrium point, a limit cycle, and torus, it is possible to analyze the attractor only by the qualitative method. The strange attractor, however, is constructed by the reconstruction, it is impossible to determine whether the attractor is constructed from a noise or a meaningful information only by the qualitative process. Therefore, the strange attractor is analyzed by analyzing the quantitative feature of the attractor.
As described above, there are various methods in analyzing the quantitative feature of the reconstructed strange attractor such as the procedure for calculating the capacity of the strange attractor, the procedure for gaining a information dimension, and the procedure for gaining a correlation dimension and the like.
The process for calculating the capacity of the reconstructed strange attractor will be described hereinafter.
Assuming that the reconstructed strange attractor is covered with a volume element such as a sphere or a hexahedron with a radius r, and that the number of the volume element which is necessary to cover the entire attractor is N(r), the relation N(r)=kr.sup.D is satisfied.
In case that the radius(r) is reduced enough, then n(r) is solved with regard to the D, the capacity of the attractor (D.sub.cap) is satisfied with the following equation. ##EQU1##
The process for gaining capacity of the attractor is performed by using the space, but it does not use the information accompanying the state variation of the given dynamics system.
Namely, the information dimension employs the following equation in analyzing the quantitative feature of the attractor which is reconstructed by using the information accompanying the state variation of the dynamics system. ##EQU2##
The P.sub.i stands for a probability in which the trace enters into the nth volume element, and .delta.(r) stands for an entropy of the dynamics system.
In the meanwhile, the most convenient procedure for analyzing the quantitative feature of the reconstructed attractor is to gain a correlation dimension, the procedure will be described with detail hereinafter.
It is gained that the number of the states in the circle having a radius (Ri) which corresponds to the distance between a state(Xi) and other state(Xj). The gained number of the states is divided by the square number of the state value (N) of the attractor, that is N.sup.2. The whole number (N) is approached to a infinite, then the correlation sum of a state(X1) can be obtained.
That is, correlation sum (C(R))=lim 1/N.sup.2 {the number of state(Xi, Xj) such that .parallel.X.sub.i -X.sub.j .parallel.&lt;R}.
With the above calculated correlation sum (C(R)), the correlation dimension (Dc) can be calculated by using the following equation. ##EQU3##
The correlation dimension (Dc) stands for a slope of the linear part of the graph of the correlation sum (C(R)) which is calculated by the equation (1).
Namely, the graph of the correlation sum (C(R)) calculated by the equation (1) is plotted in a form convergent toward a certain value. However, the attractor which is constructed by the noise is plotted in a divergent form.
And, the correlation dimension (Dc) of the attractor is obtained from the slope of the linear part of the graph which is gained by the equation (1).
As described above, in the analyzing process for a quantitative feature of the reconstructed strange attractor, the circuit seeking the correlation dimension is called a chaos processor.
In the prior art, the chaos processor has had a great large amount of calculation works, in that the correlation sum is to be obtained from the whole states (X1, X2, . . . , Xn). Namely, in order to get a correlation sum, the calculation which is equivalent to the square number of the whole states (N.sup.2 =N(N-1)/2) is needed. Particularly, the more the number of the state value, a common computer does not process the matter in a reasonable time.